Michael Welschenbach works for SRC Security Research & Consulting GmbH in Bonn, Germany. He graduated with a master's degree in mathematics from the University of Cologne, and has extensive experience in both pure and applied cryptological research. Currently, his favorite programming languages are C and C++. When not working, he enjoys spending time with his wife and two sons, programming, reading, and playing music.

• Cryptography in C and C++ MICHAEL WELSCHENBACH Translated by DAVID KRAMER ApressTM Cryptography in C and C++ Copyright c 2005 byMichaelWelschenbach Translator and Compositor: David Kramer Editorial Board: Steve Anglin, Dan Appleman, Ewan Buckingham, Gary Cornell, Tony Davis, Jason Gilmore, Jonathan Hassell, ChrisMills, Dominic Shakeshaft, Jim Sumser Assistant Publisher: GraceWong CopyManager: Nicole LeClerc ProductionManager: Kari Brooks-Copony Proofreader: Anne Friedman TEX Support: Fred Bartlett and Arthur Ogawa ManufacturingManager: Tom Debolski Cover Designer: Kurt Krames Library of Congress Cataloging-in-Publication Data Welschenbach,Michael. [Kryptographie in C und C++. English] Cryptography in C and C++ /MichaelWelschenbach; translated by David Kramer. 2nd American ed., rev. and enl. p. cm. The first American edition is a translation of the second German edition, which has been revised and expanded from the first German edition. Includes bibliographical references and index. ISBN 1-59059-502-5 1. Computer security. 2. Cryptography. 3. C (Computer program language) 4. C++ (Computer program language) I. Title. QA76.9.A25W4313 2005 005.8 dc22 2005002553 All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without the prior written permission of the copyright owner and the publisher. Printed and bound in the United States of America 9 8 7 6 5 4 3 2 1 Trademarked names may appear in this book. Rather than use a trademark symbol with every occurrence of a trademarked name, we use the names only in an editorial fashion and to the benefit of the trademark owner, with no intention of infringement of the trademark. 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Contents Foreword xiii About the Author xv About the Translator xvi Preface to the Second American Edition xvii Preface to the First American Edition xix Preface to the First German Edition xxiii I Arithmetic and Number Theory in C 1 1 Introduction 3 2 Number Formats: The Representation of Large Numbers in C 13 3 Interface Semantics 19 4 The Fundamental Operations 23 4.1 Addition and Subtraction ... 24 4.2 Multiplication ... 33 4.2.1 TheGradeSchoolMethod ... 34 4.2.2 Squaring Is Faster ... 40 4.2.3 Do Things Go Better with Karatsuba? ... 45 4.3 DivisionwithRemainder ... 50 5 Modular Arithmetic: Calculating with Residue Classes 67 6 Where All RoadsMeet:Modular Exponentiation 81 6.1 FirstApproaches ... 81 6.2 M-aryExponentiation ... 86 6.3 AdditionChainsandWindows ... 101 6.4 Montgomery Reduction and Exponentiation ... 106 6.5 Cryptographic Application of Exponentiation ... 118 v Contents 7 Bitwise and Logical Functions 125 7.1 ShiftOperations ... 125 7.2 All or Nothing: Bitwise Relations ... 131 7.3 Direct Access to Individual Binary Digits ... 137 7.4 ComparisonOperators ... 140 8 Input, Output, Assignment, Conversion 145 9 Dynamic Registers 157 10 Basic Number-Theoretic Functions 167 10.1 Greatest Common Divisor ... 168 10.2 Multiplicative Inverse in Residue Class Rings ... 175 10.3 Roots and Logarithms ... 183 10.4 Square Roots in Residue Class Rings ... 191 10.4.1 TheJacobiSymbol ... 192 10.4.2 Square RootsModulo pk ... 198 10.4.3 Square RootsModulo n ... 203 10.4.4 Cryptography with Quadratic Residues ... 211 10.5 APrimalityTest ... 214 11 Rijndael: A Successor to the Data Encryption Standard 237 11.1 ArithmeticwithPolynomials ... 239 11.2 TheRijndaelAlgorithm ... 244 11.3 CalculatingtheRoundKey ... 247 11.4 TheS-Box ... 248 11.5 The ShiftRowsTransformation ... 249 11.6 The MixColumnsTransformation ... 250 11.7 The AddRoundKeyStep ... 252 11.8 Encryption as a Complete Process ... 253 11.9 Decryption ... 256 11.10 Performance ... 259 11.11 ModesofOperation ... 260 12 Large Random Numbers 261 12.1 ASimpleRandomNumberGenerator ... 265 12.2 Cryptographic Random Number Generators ... 268 12.2.1 The Generation of Start Values ... 269 12.2.2 The BBS Random Number Generator ... 273 12.2.3 TheAESGenerator ... 279 12.2.4 The RMDSHA-1 Generator ... 283 vi Contents 12.3 QualityTesting ... 286 12.3.1 Chi-SquaredTest ... 287 12.3.2 Monobit Test ... 289 12.3.3 PokerTest ... 289 12.3.4 RunsTest ... 289 12.3.5 LongrunsTest ... 289 12.3.6 AutocorrelationTest ... 290 12.3.7 Quality of the FLINT/C Random Number Generators ... 290 12.4 MoreComplexFunctions ... 291 13 Strategies for Testing LINT 305 13.1 Static Analysis ... 307 13.2 Run-Time Tests ... 309 II Arithmetic in C++ with the Class LINT 317 14 Let C++ Simplify Your Life 319 14.1 Not a Public Affair: The Representation of Numbers in LINT ... 324 14.2 Constructors ... 325 14.3 OverloadedOperators ... 329 15 The LINTPublic Interface:Members and Friends 337 15.1 Arithmetic ... 337 15.2 NumberTheory ... 347 15.3 Stream I/O of LINTObjects ... 352 15.3.1 Formatted Output of LINTObjects ... 353 15.3.2 Manipulators ... 360 15.3.3 File I/O for LINTObjects ... 362 16 Error Handling 367 16.1 (Don t) Panic ... 367 16.2 User-Defined Error Handling ... 369 16.3 LINTExceptions ... 370 17 An Application Example: The RSA Cryptosystem 377 17.1 Asymmetric Cryptosystems ... 378 17.2 TheRSAAlgorithm ... 380 17.3 DigitalRSASignatures ... 395 17.4 RSA Classes in C++ ... 403 18 Do It Yourself: Test LINT 413 vii Contents 19 Approaches for Further Extensions 417 III Appendices 419 A Directory of C Functions 421 A.1 Input/Output, Assignment, Conversions, Comparisons ... 421 A.2 Basic Calculations ... 422 A.3 ModularArithmetic ... 423 A.4 BitwiseOperations ... 425 A.5 Number-Theoretic Functions ... 426 A.6 Generation of Pseudorandom Numbers ... 427 A.7 RegisterManagement ... 431 B Directory of C++ Functions 433 B.1 Input/Output, Conversion, Comparison:Member Functions ... 433 B.2 Input/Output, Conversion, Comparison: Friend Functions ... 436 B.3 Basic Operations:Member Functions ... 438 B.4 Basic Operations: Friend Functions ... 439 B.5 Modular Arithmetic:Member Functions ... 440 B.6 Modular Arithmetic: Friend Functions ... 442 B.7 Bitwise Operations:Member Functions ... 443 B.8 Bitwise Operations: Friend Functions ... 444 B.9 Number-TheoreticMember Functions ... 445 B.10 Number-Theoretic Friend Functions ... 446 B.11 Generation of Pseudorandom Numbers ... 450 B.12 Miscellaneous Functions ... 450 C Macros 451 C.1 ErrorCodesandStatusValues ... 451 C.2 AdditionalConstants ... 451 C.3 Macros with Parameters ... 453 D Calculation Times 459 E Notation 461 F Arithmetic and Number-Theoretic Packages 463 References 465 Index 473 viii

From the reviews of the second edition: "Addressing mathematical concepts, algorithms and C/C++ code is not a simple task; the author handles them well throughout the book. This book is translated from German by David Kramer. ... A well written book that addresses the intended purpose. ... A beginner or intermediate level C/C++ programmer can follow the text." (S. Terai, SIGACT News, Vol. 39 (1), 2008)